1. Field of the Invention:
This invention relates to position-fix navigation systems and more particularly to calculating target position and velocity from sensor information with reduced mathematical variance at times the sensors are nearly collinear with the target, a condition known as the geometric dilution of precision (GDOP).
2. Description of the Prior Art:
All position-location navigation schemes can be classified as dead reckoning position-fix or combinations of both. A position, once known, can be carried forward indefinitely by keeping continuous account of the velocity of the vehicle, ship or aircraft. This process is called dead reckoning.
A position-fix navigation system, in contrast to dead reckoning, determines the position of a vehicle, ship or aircraft without reference to any former position. The simplest position fix stems from the observation of a recognizable landmark. More frequently, however, radio aids to navigation provide position information where the physical locations of the radio aids, the ground station transmitters, are known. At present, most aircraft in the world use distance measuring equipment (DME) and very high frequency omnirange (VOR) which provide position-fix information and wherein the physical location of the DME or VOR are known. Other position-fix systems are LORAN-C and Omega as well as the recently developed position-fix systems such as the Global Positioning System (GPS) and the Relative Navigation Component of the Joint Tactical Information Distribution System (JTIDS).
The aircraft position-fixing is traditionally accomplished by determining the intersection of two or more lines of position (LOP) with respect to some known reference system. These LOP's represent the intersection of three-dimensional surfaces of position with the earth's surface which in many practical cases is approximated locally by a plane. Thus, on the earth's surface, the LOP or a range measurement is a circle while an angle (bearing) measurement is a line. A position-fix is the intersection of two LOP's which require at least two measurements. Since no measurement can be made without error, the LOP's will have an error associated therewith and thus the intersection of two LOP's will generate a position error.
The position error may be substantially increased at times the LOP's are nearly collinear (i.e. not orthogonal). This position error mechanism is known as the geometric dilution of precision (GDOP). One example where the LOP's are nearly collinear is when an aircraft is between two landmarks along a line between the landmarks or when the aircraft is outside both landmarks along a line between both landmarks. Another example of GDOP is when the aircraft distance from two landmarks is great compared to the distance between the two landmarks. Position error due to GDOP begins to occur when the angle .gamma. between the intersection of two LOP's is greater that 150.degree. or less than 30.degree..
Virtually all aircraft position-fix algorithms use some form of unbiased Least Mean Squared Estimation (LMS). The position of the aircraft is expressed as a plurality of linear equations which are then solved for the unknown values of position in coordinates (X and Y) and velocity having components V.sub.x and V.sub.y. The linear equations may be expressed in the form EQU Y=X.beta.+e (1)
where Y is the nxl observation vector, X is an nxp prediction matrix and e is the nxl error vector with covariance matrix, W. The statistical problem is how to best guess the component values of the pxl regressor vector .beta.. As is well known, one method is the ordinary LMS solution which assumes the covariance matrix W as given in Equation (4). It is given in Equation 2, EQU .beta..sub.OLS =(X.sup.T X).sup.-1 X Y (2)
which follows from the so called "normal" equations, shown in Equation 3, EQU X.sup.T X.beta..sub.OLS =X.sup.T Y (3)
Equation (3) assumes that EQU W=.sigma..sup.2 I (4)
where .sigma..sup.2 is the variance (i.e. .sigma. is the standard deviation) and I is the nxn identity matrix. When ##EQU1## the LMS solution .beta..sub.GEN is called Generalized least square where EQU .beta..sub.GEN =(X.sup.T W.sup.-1 X).sup.-1 X.sup.T W.sup.-1 Y (4.2)
A most critical performance measure is the variance of the estimate; it is given by EQU VAR(.beta..sub.OLS)=(X.sup.T X).sup.-1 .sigma..sup.2 ( 4.3)
for the OLS estimate.
When GDOP exists, the sensor errors i.e. receiver noise, quantization noise, propagation effects, etc. can be "blown up" or "blown down" when the sensor measurements are referenced to the navigation coordinates to determine an aircraft position fix. This inflation of the position estimate errors (GDOP) arises mathematically from the X.sup.T X term in Equation 4.3. It may have values which are very small or values which are equal to zero in the diagonal terms of the matrix. These unusually small or zero values "blow up" to be very large values when the inverse of X.sup.T X (i.e. (X.sup.T X).sup.31 1 is calculated as indicated in Equation 4.3. The meaning of Equation 4.3 is that on the average the very large values in the matrix (X.sup.T X).sup.-1 inlate the values of the ordinary LMS solution .beta. when Equation 2 is solved.
Present navigation systems on aircraft accept the position estimates, some of which are highly inflated.
Mathematically, a plurality of linear equations may be solved wherein the relationship X.sup.T X is used where X.sup.T is the transpose of the matrix X and where X.sup.T X is a matrix having eigenvalues .lambda.. When the relationship X.sup.T X moves from a unit matrix to one where high multicollinearity exists, variance inflation will rise as the smallest eigenvalue .lambda.s approaches zero. A mathematical technique which counteracts the effects of multicollinearity was disclosed in a publication by J. Riley wherein the diagonal terms of the X.sup.T X matrix were limited to certain minimum values. This technique avoided numerical difficulties when inverting a square matrix. The publication by J. Riley is entitled "Solving Systems of Linear Equations--With a Positive Definite, Symetric But Possibly Ill-conditioned Matrix," Mathematic Tables and Other Aids to Compute., Vol. 9, 1955. In a further development, a paper was published by A. Hoerl and R. Kennard entitled "Optimum Solution of Many Variable Equations" Chemical Engineering Progress, Vol. 55, No. 11, November 1959. Two additional papers were published by A. Hoerl and R. Kennard entitled "Ridge Regression and Bias Estimation for Nonorthogonal Problems", Technometrics, Vol. 12, No. 1, February 1970 and "Ridge Regression: Applications to Nonorthogonal Problems", Technometrics, Vol. 12, No. 1, February 1970. The authors described a technique termed "Ridge Regression" which counteracts the effects of multicollinearity. In Ridge Regression, the diagonal components of a matrix are limited to a predetermined value by adding a small term K to each diagonal component. That is X.sup.T X is transformed into X.sup.T X+KI where I is the unit matrix. The idea in Ridge Regression is to find an estimator whose variance decreases as [X.sup.T X+KI].sup.-1 ; which means that the variance inflation is limited to 1/K even when the eigenvalues of X.sup.T X.fwdarw.0. Equation 5 shows the expression for the Ridge Regression coefficients .beta..sub.R. EQU .beta..sub.R =[X.sup.T X+KI].sup.-1 X.sup.T Y. (5)
In the literature, K is known as the Ridge parameter.